Chapter 4. The Organization of Instruction

Having described the mathematical content of lessons, we turn now to examine more fully the methods by which teachers structure lessons so as to engage students with the content. It would seem that without rich mathematical content, students would be unlikely to learn mathematics. Yet even with content that is of high quality, there are many options available to teachers as they begin the practical task of constructing classroom lessons. In this section we will begin to describe the kinds of lessons teachers construct in the three countries.

CHARACTERISTICS OF THE CLASSROOM

An initial viewing of the videotapes suggests that the similarities among the eighth-grade classrooms we visited were more striking than the differences. Although we were videotaping in three highly distinct cultures, there was nevertheless considerable similarity in classroom arrangement in all countries. In all three nations, classrooms typically contained one teacher and many students. All of the classrooms we visited contained chalkboards, and all contained individual desks for each student.

The arrangement of desks in the classrooms varied somewhat by country (see figure 35). In Japan, 94 percent of lessons had desks arranged in rows facing the front; the share was 77 percent in the United States and 67 percent in Germany. A substantial number of classrooms in Germany (22 percent) had desks arranged in a U-Shape configuration, with the open end of the U facing the front. (Most often, the inside of the U contained rows of desks facing the front of the room.) Some classrooms in each country had desks arranged in groups.

Figure 35

Arrangement of desks in German, Japanese, and U.S. classrooms

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

BASIC CHARACTERISTICS OF THE LESSON

Our first step in coding the videotapes was to mark the points at which the lesson itself began and ended. Videographers made a judgment about when to turn on the camera, but this was usually before the beginning of instruction proper. Similarly, the camera kept on rolling for a few minutes after the teacher ended the lesson. We needed a consistent way of coding the beginning and ending of the lesson because many of our outcome measures (e.g., duration of time devoted to some activity) are reported as a percentage of total lesson time. Obviously, these measures will be affected by the total measured duration of the lesson.

We divided each class into three segments: Pre-Lesson Activity, Lesson, and Post-Lesson Activity.

• PRE-LESSON ACTIVITY comprises all nonmathematical activity or talk that occurs before the lesson proper begins. Examples of such activity include greetings, teacher announcements about extracurricular activities, changes in school schedule, past and future exams, or housekeeping. In addition, talking about previously assigned homework in general terms, such as grades, points, etc., without an elaborated explanation would also be part of PRE-LESSON ACTIVITIES;
• LESSON is defined as the period of time that is allocated by the teacher for mathematical-instructional activities. The teacher generally indicates the beginning of the lesson with an explicit verbal marker, such as "OK, today we are going to..." Likewise, the end of the lesson is marked as well. Often there are bells that coincide with the beginning and end of the lesson; and
• POST-LESSON ACTIVITY is defined as all nonmathematical activities that occur after the end of the LESSON and before the end of CLASS. These segments include teacher announcements about homework without elaboration, directions to clean up desks and chairs, and other housekeeping activities.

The average duration of the lessons in our sample (subtracting out time devoted to pre- and post-lesson activities) was 43.2 minutes in Germany, 49.5 minutes in Japan, and 49.4 minutes in the United States. German lessons were significantly shorter than those in Japan or the United States. In the United States the shortest lesson was 32.0 minutes, the longest, 91.2 minutes. The range in Germany was 34.2 to 49.4 minutes, and in Japan, 43.4 to 54.9 minutes.¹

Figure 36

Percentage of lessons with at least one outside interruption

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

In addition to the starting and ending points of the lesson, we also coded whenever there were interruptions from outside the classroom during the lesson. This included such events as announcements over the public address system or visitors who interrupted the lesson. These results are presented in figure 36. Such interruptions were never observed during the Japanese lessons but were relatively common during the U.S. lessons. Interruptions were significantly more likely to occur in U.S. and German lessons than in Japanese lessons.

ORGANIZATION OF THE LESSON

Classwork and Seatwork

Having marked the beginning and end of each lesson, our next step was to divide the lessons into organizational segments. Although in many respects lessons look quite different across the three cultures, teachers everywhere tend to divide their lessons into periods of classwork and periods of seatwork, and it is not difficult to reliably code the beginnings and ends of these segments. We identified three distinct kinds of segments:

• CLASSWORK segments (coded as CW) are defined as those times when the teacher is working with all (or most) of the students in a whole-class situation; the type of talk is predominantly public, that is, the audience is the whole class. Possible activities during CLASSWORK include the teacher and students engaged in learning a new concept, reviewing a previously learned concept, solving mathematical problems together, or sharing solutions to problems they are solving;
• SEATWORK (coded as SW) is defined as a period of time during the lesson when students work independently on assigned tasks, either alone or in small groups (two students or more). The type of talk is predominantly private, but there are instances of public talk when the teacher makes announcements to the whole class. The beginning of SEATWORK is usually marked by one or more of the following: (1) The teacher announces that students should begin their independent work; (2) There is a period of silence after the teacher provides necessary information to students, or (3) students actually start working; and
• CLASSWORK/SEATWORK COMBINATION segments are the rare cases in which the teacher assigns some students to work independently on a particular task and situation, while the rest of the class works with the teacher. In such cases, the Organization of Interaction is coded as CW/SW.

Organizational segments were coded exhaustively, meaning that the end of one segment was the beginning of the next. Seatwork segments were further characterized as being individual, group, or both. During individual seatwork segments students worked independently, by themselves; during group seatwork segments they worked in groups.

Results indicate that Japanese lessons contain considerably more organizational segments on average than do German or U.S. lessons (figure 37).

Figure 37

Average number of organizational segments in German, Japanese, and U.S. lessons

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

As it turned out, very few of the segments in any country were coded as classwork/seatwork combinations, and these segments accounted for less than 1.5 percent of lesson time, on average. The bulk of the segments were either classwork or seatwork. Figure 38 shows the average number of classwork and seatwork segments per lesson in each country. Again, Japanese lessons contain the highest average number of segments of both types. There was no significant difference between U.S. and German lessons in this regard.

Figure 38

Average number of classwork and seatwork segments per lesson in each country

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The picture changes somewhat if we look at the percentage of time during the lesson that is spent in classwork and seatwork (figure 39). Here, Japan and the United States look quite similar, and Germany looks different. Japanese and U.S. teachers spend almost identical percentages of time in classwork (approximately 60 percent) and in seatwork (approximately 40 percent). German teachers spend a higher percentage of their lesson time in classwork (about 70 percent) than do teachers in the other two countries. Conversely, Japanese and U.S. teachers spend a higher percentage of their lesson time in seatwork than do German teachers.

Figure 39

Average percentage of time during the lesson spent in classwork and seatwork in each country

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

While Japanese lessons have more organizational segments, the length of each segment is shorter in Japan than it is in Germany or the United States. Figure 40 shows the average duration in minutes of CLASSWORK and SEATWORK segments in each country. The average duration of SEATWORK segments in U.S. classrooms was significantly longer than in either German or Japanese classrooms. The average duration of CLASSWORK segments were significantly shorter in Japan than in either Germany or the United States.

Figure 40

Mean duration of classwork and seatwork segments in each country

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Moreover, analysis of the data shows that students in German classrooms spent a higher percentage of time working individually during SEATWORK than did students in Japanese classrooms (figure 41). Students in Japan spent a higher percentage of time working in groups during SEATWORK than students in Germany or the United States.

Figure 41

Percentage of seatwork time spent working individually, in groups, or in a mixture of individuals and groups

 

NOTE: Percentages may not sum to 100.0 due to rounding.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We also looked at patterns of seatwork within the lesson. In figure 42 we show the percentage of lessons in each country that include individual seatwork, group seatwork, both kinds of seatwork, or no seatwork. Although there was no significant difference across countries in the percentage of lessons containing only groupwork, a greater percentage of Japanese lessons contained both group and individual seatwork than did the German or U.S. lessons. German classrooms were more likely to have no seatwork than the Japanese or U.S. classrooms. Both German and U.S. classrooms contained a higher percentage of individual-only seatwork than Japanese classrooms.

Figure 42

Percentage of lessons in each country in which seatwork of various kinds occurred

 

NOTE: Percentages may not sum to 100 due to rounding.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

To summarize, Japanese teachers tend to change more frequently from classwork to seatwork and back again within the lesson than do German and U.S. teachers. And, they tend to alternate between seatwork segments where students work individually and those where students work in small groups. German teachers concentrate on classwork, and provide less variation in organization of interaction over the course of a lesson than do either Japanese or U.S. teachers. U.S. teachers resemble Japanese teachers in terms of the time they devote to seatwork, but look more like German teachers in terms of the less frequent change between classwork and seatwork.

Activity Segments

Having coded lessons into segments of classwork and seatwork, we now proceed to the next layer. Classwork and seatwork, after all, represent only the most superficial view of what happens in a mathematics lesson. What goes on during these segments, and what goals are teachers trying to achieve? Our next step was to further divide the lesson into activity segments.

What we call activity segments are segments of the lesson that serve some specific pedagogical function. Examples of such functions would be setting up for seatwork (i.e., getting students ready to work on their own), working on tasks, or sharing the results of seatwork. These kinds of activities appear in all cultures and can be defined in a cross-culturally valid way. They also are categories that map well onto teachers' views of how lessons are planned and implemented. In fact, teachers generally mark the transitions between these activities with explicit words and actions. For example, a teacher might say, "Everyone get in your groups and do the problem I've written on the board." This marks a clear shift in activity, as well as a shift from classwork to seatwork.

We defined four major categories of activities: SETTING UP, WORKING ON, SHARING, and TEACHER TALK/DEMONSTRATION. The goal of SETTING UP segments is to prepare students for a subsequent seatwork segment. SETTING UP situations occur when the teacher assigns the task(s) and/or situation(s) for students to work on independently during seatwork. We identified two subtypes of SETTING UP segments:

• SETTING UP: MATHEMATICAL was coded when the teacher presented task(s) and/or situation(s) to the students with explanations or discussion; and
• SETTING UP: PHYSICAL/DIRECTIONAL was coded when the teacher presented students with task/situations without additional explanations or discussions. These segments usually included physical activities, such as moving into groups, passing out handouts, writing down task(s) and/or situation(s), and/or directions.

WORKING ON segments were the most common. Although WORKING ON was most commonly coded during periods of SEATWORK, it also could be coded during CLASSWORK. During CLASSWORK segments, WORKING ON occurred whenever the teacher and the students worked collaboratively on task(s) and/or situation(s), or derived/learned principles, properties, or definitions (PPDs). When the shift between SETTING UP and WORKING ON was not clearly identified, SETTING UP was included in the WORKING ON segment. We coded four types of WORKING ON segments:

• WORKING ON TASK/SITUATION was coded whenever the teacher and/or students worked on tasks and situations not included in the following three categories;
• WORKING ON HOMEWORK. This included segments in which homework was assigned but not necessarily started;
• WORKING ON TEST; and
• WORKING ON MULTIPLE ACTIVITIES was coded when the students were engaged in two or more assignments, such as checking homework answers and starting on a worksheet.

We coded SHARING segments when the activity focused on presenting, discussing, and reflecting on previously completed tasks and situations. The results might be shared in the form of teacher presentation, student presentation, interactive discussion, or visual representations. The segment began when the teacher expressed the intention of sharing the produced results, and it ended when there was a shift in activity that normally occurred in conjunction with a shift in content.

There were three kinds of SHARING segments, depending on what was being shared:

• SHARING TASK/SITUATION;
• SHARING HOMEWORK; and
• SHARING TEST.

TEACHER TALK/DEMONSTRATION was coded when the teacher talked about concepts, ideas, solution strategies or methods, lesson goals, or demonstrated solution steps. This segment was limited to situations where the teacher was transmitting lesson-relevant information to the students, and the students' role was to listen. Students could ask questions of the teacher, and the teacher could ask questions if his/her primary purpose was to maintain the students' attention. For example, if the teacher lectured but paused every 5 minutes to say, "OK, got it?" this would still be coded as TEACHER TALK/DEMONSTRATION, provided the responses were limited to a simple nodding of heads or murmuring of "Uh huhs." If the teacher elicited responses from the students the segment was coded as WORKING ON TASK/SITUATION. Instances where the teacher sets up a task and situation or comments on students' solution methods in the form of a presentation do not fall into this category but into WORKING ON or SHARING.

Finally, OTHER was coded if the content of the segment was not related to the mathematical content of the lesson. These segments may include instances when the teacher checks off that students completed their homework, small talk, housekeeping, or discipline. If a segment contained two or more types of activity occurring simultaneously, we coded it as MIXED.

The resulting 12 categories of activity segments are listed in figure 43.

Figure 43

Overview of categories for coding lesson activity segments

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

As can be seen in figure 44, Japanese lessons contained significantly more activity segments on average than lessons in Germany. Thus, the more frequent changing from classwork to seatwork in Japan is accompanied by a more frequent changing from one activity to another.

Figure 44

Mean number of activity segments in German, Japanese, and U.S. lessons

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We turn now to a discussion of the time devoted to different activities within each culture. Three of the codes--WORKING ON TEST, WORKING ON MULTIPLE ACTIVITIES, and MIXED--each accounted for less than one-half percent of lesson time. Thus, we will not present any further data concerning these codes. On the other hand, some interesting differences emerged when we examined the distribution of the various activity types across cultures. We will present some of these differences next.

Time Spent in Other Activity

We start by looking at the time devoted to OTHER activities during the mathematics lesson. OTHER was coded whenever the activity was unrelated to mathematics or to the current lesson. For example, a teacher might pause during the lesson to discuss a recent field trip or sporting event of interest to the students.

This kind of unrelated activity was more common in the United States than in either of the other two countries. In figure 45 we show two different measures of OTHER: as a percentage of total time (panel a) and as a percentage of lessons in which OTHER was coded (panel b). Using percentage of total time as the measure, U.S. lessons devoted significantly more time to unrelated activities during the lesson than did either German or Japanese teachers. Similarly, a significantly higher percentage of U.S. lessons contained such activities than did German lessons. Although the total percentage of time devoted to unrelated activities during the lesson is small in all countries, even a brief diversion of this sort may break the flow of the lesson. Thus, it may be important that nearly 25 percent of the U.S. lessons included this kind of activity.

Figure 45

Time devoted to unrelated activities during the mathematics lesson: (a) as a percentage of total lesson time and (b) as percentage of lessons in which any activity is coded as "other"

(a)	(b)

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Homework During the Lesson

There is a great deal of interest in the United States in the topic of homework: Many Americans believe that more homework will translate into higher achievement for students. In fact, our observations confirm that homework plays a more prominent role in eighth-grade mathematics classrooms in the United States than it does in Japan, and somewhat more prominent than in Germany.

In figure 46 we show the percentage of lessons in which students actually work on or share homework. Japanese students were never observed to work on the next day's homework during class, and were relatively rarely observed to share homework results. Both German and U.S. students share homework in class frequently, but only U.S. students spend significant amounts of time in class actually working on the next day's homework. When we look at total percentage of time during the lesson devoted to assigning, working on, or sharing homework, we get a similar result: Two percent of lesson time in Japan involves homework in any way which is significantly less than the 8 percent of lesson time in Germany and 11 percent in the United States.²

Figure 46

Percentage of lessons in which class works on and shares homework (not including assigning homework)

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

In our teacher questionnaire we asked whether or not they had previously assigned homework that was due on the day of the videotaped lesson. The countries differed in the proportion that had done so. In 55 percent of both the U.S. lessons and the German lessons, the teachers said that they had assigned such homework, compared to 14 percent of Japanese teachers who responded in this way.³

Teacher Talk/Demonstration

TEACHER TALK/DEMONSTRATION was coded when teachers engaged in lecturing. Here teachers would simply talk, with or without objects, presenting lesson-relevant information to students.

As can be seen in figure 47, there was more TEACHER TALK/DEMONSTRATION in Japan than in the other two countries. As with OTHER, the overall percentage of time devoted to this activity was not large (see panel a), but more Japanese lessons included at least some TEACHER TALK/DEMONSTRATION than did those in either Germany or the United States (panel b).

Figure 47

Emphasis on teacher talk/demonstration as indicated by (a) percentage of lesson time, and (b) percentage of lessons in which such segments occur

(a)	(b)

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Working On Tasks and Situations

Much of the instructional time in all countries occurs during three types of segments: SETTING UP tasks and situations, WORKING ON tasks and situations, and SHARING of tasks and situations. SETTING UP and SHARING occur during classwork; WORKING ON tasks and situations can occur during either classwork or seatwork.

By far the most frequent and time-consuming type of activity segment in all countries is WORKING ON TASK/SITUATION. As can be seen in panel (a) of figure 48, classrooms in all countries spent 60 percent or more of lesson time in WORKING ON TASK/SITUATION segments. German classrooms, however, spent more time this way than Japan. Consistent with our findings on classwork and seatwork, Japanese WORKING ON TASK/SITUATION segments were significantly shorter in duration on average than were such segments in Germany and the United States (panel b).

Figure 48

(a) Percentage of total lesson time spent in and (b) average duration of working on task/situation segments

(a)	(b)

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

A more detailed picture emerges when we break down WORKING ON TASK/SITUATION according to whether it occurs during classwork or during seatwork. As seen in figure 49, German classes spend more time WORKING ON tasks and situations during classwork than during seatwork; Japanese classes spend more time WORKING ON tasks and situations during seatwork than during classwork. U.S. classes spend roughly equal time WORKING ON tasks and situations during classwork as they do during seatwork. Moreover, German classes spend more time WORKING ON tasks and situations during classwork than either their Japanese or U.S. counterparts.

Figure 49

Average percentage of lesson time spent in (a) working on task/situation during classwork, and (b) working on task/situation during seatwork

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Setting Up and Sharing Tasks and Situations

Finally, although Japanese classes spend relatively less time WORKING ON tasks and situations during classwork, they spend more time SETTING UP and SHARING tasks and situations than do both German and U.S. classes, as seen in figure 50.

Figure 50

Average percentage of total lesson time spent in setting up and sharing task/situation

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We have examined the content of the lessons and the ways that teachers in the three countries organize their lessons. We now present some preliminary analyses of the processes that go on as lessons unfold.

¹ Standard errors of the average duration of lessons in Germany, Japan, and the United States were 0.24, 0.35, and 1.92, respectively.

² Standard errors for Germany, Japan, and the United States were 1.75, 0.46, and 1.89, respectively.

³ Standard errors for Germany, Japan, and the United States were 4.86, 5.46, and 7.12, respectively.